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Current time:0:00Total duration:7:56

Video transcript

a line passes through the points negative three six and six zero find the equation of this line in point-slope form slope-intercept form standard form and the way to think about these these are just three different ways of writing the same equation so if you give me one of them we can manipulate it to get any of the other ones but just so you know what these are point-slope form let's say that X let's say the point x1y1 are it let's say that that is a point on the line and when someone puts this little subscript here so if they just write an X that means that we're talking about a variable that could take on any value if someone writes X with a subscript 1 and a y with a subscript 1 that's like saying a particular value X a particular value of x and a particular value of y or particular coordinate so and you'll see that when we do the example but point-slope form says that look if I know a particular point and if I know the slope of the line then putting that line in point-slope form would be Y minus y1 is equal to M times X minus x1 so for example and we'll see do that in this in this video if the point negative 3 comma 6 is on the line then we'd say Y minus 6 is equal to M times X minus negative 3 so it'll end up becoming X plus 3 so this is a particular X and a particular Y it could be a negative 3 & 6 so that's point slope form slope intercept form is y is equal to MX plus B where once again M is the slope B is the y-intercept where does the line intersect the y axis what value does y take on when x is 0 and then standard form is the form ax plus B y is equal to C where these are just two numbers essentially they really don't have any interpretation directly on the graph so let's do this let's figure out all of these forms so the first thing we want to do is figure out the slope once we figure out the slope then point-slope form is actually very very very straightforward to calculate so just to remind ourselves slope slope which is equal to M which is going to be equal to the change in Y over the change in X now what is the change in Y if we view this as our endpoint if we view this if we imagine that we're going from here to that point what is the change in Y well we have our endpoint which is zero y ends up at zero and why was it six so they are finishing Y point is zero our starting Y point is six what was our finishing x point or x-coordinate our finishing x-coordinate was six let me make this very clear I don't want to confuse you so this zero we have that zero that is that zero right there and then we have this six which was our starting Y point that is that six right there and then we want our finishing x-value that is that six right there or that six right there and we want to subtract from that we want to subtract from that our starting x-value well our starting x-value is that right over there that's that negative three that's that negative three and just to make sure we know what we're doing this negative three is that negative three right there I'm just saying how much did if we go from that point to that point our Y went down by six right we went from six to zero ry-ry went down by six so we get zero minus 6 is negative six that makes sense why went down by six and if we went from that point to that point what happened to X we went from negative 3 to 6 it should go up by 9 and if you calculate this take your six minus negative three six minus negative three that's the same thing as 6 plus three that is nine and what is negative 6 9 so well if you simplify it it is negative 2/3 you divide the numerator and the denominator by 3 so that is our slope negative 2/3 so we're pretty much ready to use point-slope form we have a point we could pick eat one of these points I'll just go with the negative 3 6 and we have our slope so let's put it in point-slope form so point point-slope form point-slope form all we have to do is we say why - now we could have taken either of these points I'll take this one so Y minus the Y value over here so Y minus 6 is equal to is equal to our slope which is negative 2/3 negative 2/3 x times X minus our x-coordinate well our x-coordinate so X minus our x-coordinate is negative 3x minus negative 3 and we're done we can simplify a little bit this becomes Y minus 6 is equal to negative 2/3 times X X minus negative 3 is the same thing as X plus 3 this is our point-slope form now we can literally just algebraically manipulate this guy right here to put it into our slope intercept form let's do that so let's do slope intercept in orange so we have slope slope intercept slope intercept so what can we do here to simplify this well we can we can multiply out the negative 2/3 so you get Y minus 6 is equal to I'm just distributing the negative 2/3 so negative 2/3 times X is negative 2/3 X and then negative 2/3 times 3 is negative 2 and now to get it in slope-intercept form we just have to add the 6 to both sides so it gets so we get rid of it on the left hand side so let's add 6 to both sides of this equation left-hand side of the equation we're just left with the Y these guys cancel out you get a Y is equal to negative 2/3 X negative 2/3 X negative 2 plus 6 is plus 4 so there you have it that is our slope-intercept form MX plus B that's our y-intercept now the last thing we need to do is get it into the standard form get it into standard form standard so once again we just have to algebraically manipulate it so that the X's and the Y's are both on this side of the equation so let's just add 2/3 X to both sides of this equation so let me well I'll start here so we have y is equal to negative 2/3 X plus 4 that's slope-intercept form let's add 2/3 X so plus 2/3 X plus 2/3 X to both sides of this equation plus 2/3 X I'm doing that so I don't have this 2/3 X on the right hand side this negative 2/3 X so the left hand side of the equation I scrunch it up a little bit maybe more than I should have the left hand side of this equation is what it is 2/3 X because 2 over 3x plus this Y that's my left-hand side is equal to these guys cancel out is equal to 4 so this by itself we are in standard form this is the standard form of the equation if we wanted to look make it look extra clean put no have no fractions here we could multiply both sides of this equation by 3 if we do that what do we get 2/3 x times 3 is just 2 X Y times 3 is 3 y and then 4 times 3 is 12 these are the same equations I just multiplied every term by 3 if you do it to the left hand side you can do it to the right hand side or you have to do to the right hand side and we are in standard form